Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis
Number Theory
2025-07-10 v1
Abstract
In an earlier paper, we proved that Montgomery's Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function can be used to prove without the assumption of the Riemann Hypothesis (RH) that asymptotically 100% of the zeros are both simple and on the critical line. This is based on a method of Gallagher and Mueller from 1978. We formulate an appropriate form of the Alternative Hypothesis (AH), which determines a different PCC, and, using the same method as above, prove that asymptotically, 100% of the zeros are both simple and on the critical line. As in our previous paper, we do not assume RH.
Keywords
Cite
@article{arxiv.2507.06823,
title = {Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis},
author = {Daniel A. Goldston and Junghun Lee and Jordan Schettler and Ade Irma Suriajaya},
journal= {arXiv preprint arXiv:2507.06823},
year = {2025}
}
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11 pages